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Deactivation of He 23S by thermal electrons
.’
C!H?3WCAt PHY!XiS
volume 25;numbkQ
I_ETi’EiS
:
15 April 1974
._
DEACTIVATION OF He 2% BY THERMAL ELECTRONS RX. NEiBE’k, R.S. CIBEROI* I&VResearch Labomtory, San Jose. CahfOna 95193. US4 and
P/~ysicsDepartment. University
J.N. BARDSLEY of Pittsburgh,Pittsburglt. Pemtsyhwtia 15213, US.4
Received 7 February 1974
The rate coefficient for deactivation of the metastable 23S state of He by impact of thermal electrons is deduced from recent cakulations of inefastic electron-He cross sections. The deactivation rate is found to be nearly constant with tethperature. Computed values range between 2.82 and 3.03
Cross sections
for electron
impact
excitation
of
Table 1 Total computed deactivation cross section Qzt for H=z$~~S-* 1 ‘S) in units no: and k, Qa, in units nao for dent eIectron wavevector k2(uij*)
the n = 2 states of He have recently been computed by a variational method [I]. The variational method is equivaIent to solution of continuum Bethe-Goldstone equations for the coupled open scattering channels resulting from target atom states lrS, 23S, 2’5, 23P, 2tP, including the most important effects’ of poia~ation or virtual excitation of the n = 2 electrans. The computed 1’S + ?3S excitation cross section is in excellent agreement.with the experimental crosssection f2], if the arbitrary absolute norm&zation of the latter is adjusted to the computed cross section. In view of this agreement,.the corresponding de+xzt%arrbncross sectiop~hould be given to the_ use-’ ful accuracy by the variagorial caIcu!atitins. T&e purpose of this note is to present this d&a, which cann& be.deduced’accurateIy from the graphs published in ref. $ I], and t& derive from it the rate con&ant KB,(2, I) for .deactiv&tion of He(2%) by thermal.- .. electrons. By d&i&d balance, the dea&iv&ion cross &&on
k2
O-02
7.968
0.1594
0.14 0.18 0.22
O-i385 0.1542 0.1763 0.1672 0.1310
0.26 Q.30 0.34
0.319 0.282 -0.180
0.0828 0.0845 0.0611
0.38 0.42
0.x45 0.1 I?
0.0493
Q;; =
;
0.10
-.
kzQzt
2.308 1.542 1.260 0.929 ‘_. 0596
0.06
..
Q21
inci-
0.055 I
0)
where, in Hartree atomic units, the electron energy in the 11s channel iiki, and the energy in the 23S
channcI~is&k$.For low-energy eIeetrons in the second channel, analysis of threshold behavior (3 indicates by, :.:.:.. ‘that Qi;? should va& as kz andQ21 as kg 1 , and that . .._’ : -..._., ._-., ’ the Cross s&+ons are dqminated by the s-wave (2s ‘_ _ . 2.gcatt&pg state&.tiomputed values of Qzr are shown ’ *‘s’&or@d’in pa by-t% dffioe of Na& R&ear& &es : ‘.. .. : in table 1..,Asexpected;&2 Q2i is nearly constant, alC+rtict No_~NOOi&720051; -. .. : ‘. . .; _ ‘. Qzl. is related t+the 2% excitation cross sectiWQ12-
‘. .,. .
-.
;
..-, ..
1 . .
-. _.:
” I ‘:,
-.
.’
_.
._‘..“.
.*
;
,.
.:
:.
., .
.
.,
-‘-
_,.
.‘.
.
,-.
.”
‘..I
,:::-
*
.I
.;
.
_.. ,?. ,_ .-.’::,
. .._,.- _. .. _ -’ __ : ..;:
,, ., ., ,__.. .::-.::.
-..‘.
. .
1, :-:.; ‘; _:_:; _::__. -;.:, _’ _:..:‘-:_.. .. ._
587 _ ,c.’ .,. ( ,. -_.
:
CHEMIQIL PHYSICS LETTERS though
the zP contribution,
which increases
with-k:,
Tablei Deactivation rate coefficient for He(23S -, I’S)
is significant for k2 > 0.10 a&‘. The % contribution to k2 = 0.22 a;’ is less than lO%, and higher partial waves have not been included in the calculations. The deactivation rate coefficient KH,(2, 1) is the .thermal average of u& , where u is the electron Velocity. In atomic units (n(yca%), this is just the thermal average of k, QZ1, K(2,l)
=
Jr exp(-k2/2kg
T)k2dk.
’
42 379 1053
(2)
variable
t* = k;/2kB T _ is
k2Q21 =g(t)=g,,
+g2t2 +g4t4 +.__,
e-r%2n+2dt
= +r(n
+$)
=g0’+3g2
+ ag4
(6)
For numerical values ofg(t), eq. (2) can be evaluated by Gauss quadratilre, using t2ePt2 as the density function in the integrals. For an IV-point quadrature, the weights toi and abscissae ti are to be chosen so that
T= k;/3kB
F. wi8ttf> ‘
(7)
gives the exact value of K(2,l)
for any polynomial including tkms up to g4N_2 _The level of approximation. that neglects terms beyond g2 corresponds to N= 1, with : r2=3-.. w1=l, 1. 1 @I In order to:deter&ne definitive~values of the deexcitation cross section;.it.wckld be necessary tb carry put izal&lations on-a more cIos&ly spaced grid ofk2 %%s HXW thy= i+labl~
;_.,_.
.:
_’ _-..__., ..-_.-.;.,.- . . , ;
2.52
bye, yd to refine-the
_.. .~.-. ,.;. .b _._..y_.--__ ::: _:.,.-, ‘.‘ :.I._-,_. -- .; -_ ._
2.94 2.83 2.82 2.94 3.03
calculations
until k2 Q21 becomes
insensi-
tive to further improvements of the variational wavetinction. Since this has not been done in the present case, a numerical or least-squares fit to k2 Q21 may not be reliable to terms beyondg2_ If this is true, it would be illusory to use a quadrature formula more elaborate than the one-point formula with constants given by eq. (8)_ Since WI = 1, the one-point formula simply associates each given value of k2 Qzl with K(2,l) for an effective temperature determined by eq. (3) with t2 = $. This is
(5)
+ .. . .
5095
variational
and K(2,l)
2.97 3.39 3.22
(4)
the integrals required are of the form
s 0
2063 3410
125 250 500 1000 2000
(3)
If this polynomial
:
3.07 2.67
Table 3 Deactivation rate coefficient (units 10mgcm3/sec), obtained by thermal average of kQ values given in table 1
where kB is the Bokzmann constant (3.1667 X low6 au/%)_ From the values given in table 1 it appears reasonable to approximate k2 Q2i by a polynomial in k$ or in the dimensionless
in units
10-9cm3/sec
T) [kQzl ] k*dk
I,” exp(-k2/2k,
.lS A~+1974
_
The resulting values of K(2, l), for effective values of T corresponding to the entries in table 1, are listed in table % It should be noted that es._(g) is just the: &s&al equipartition law for the momentum of free electrons. However, the. a&ume@ here is riot based on’any assumption of sha~$y-determiried electron mom&turn,but rather on the relatively smooth behavior of k2 Qzl over a range qf en@rw.$ahies. tint3 more-detailed ; .: _- c?mputationi .ofk2 Qil have tieen darried out, it is ., felt *at the pre$entaGoti df yaluespf K(2,.l)+ giy& assump-._:. : -. ,.._.intable _.. 2.k t$e best way to a&id .. I, arbitnky .. .,__ :: _: .:.:
: ..-.: 1.‘.:,
j
_ .I -_ : .:. -..-.. _.,‘. _,- ._.-. -:: 7. ._ .... . : .;
,J’ ; _.,..: -;- ,:,.:~L :, .-.:. ‘._-’
., _
‘V&i&e is, number 4 ..‘. 1 :_
t&s
‘,
I
: _.
:
..
;
,’
CfiEFpUqPIWSICS
I’.-.
about tb6 functional pehavio? of ki &ii &I the
energy tinge ctinsidered her& A. -. ‘. The value$‘ofkz& iisted in table.1 ‘de&i a poly-’ -;nomial, ati in eq. (4), in@iding' teinis’up t6gzo. The i_ weights wi in the corresponding equat-int&al quadiaL ture formula a& easily obtained by inverting m i I X 1I m&ix. Values of K(2, I) cdtiputed by this quadrature formula are given in table 3.
In com~a~son with values 0fk”~o(2,1) obtained by B&es et al. [4], from e&lie1 scattering .d@a, the
_.
I&lTERS
.’
...
‘.
__
.._ .:
_.
.‘_’
.,
:’
..,‘.
:
,,
:.
._,
:
I5 April 1974
present vat&s ar~&&ficantly larger at tow tempera‘.tures, and are eisenti~y c&xtant with te@perature.
: .. ‘, .: ,: :- .‘,_, : : -, Refeiexices
:’ ,:, .,
[l] R.$, o&&i and RX. Nesbet, Phys. Rev. AS (1973) 2969. 12J H.H. Bron@rsma,F.W.E.Knoop and C. Back%Chem. Phys. Letters 13 (1972) 16.
f3] J.N. Bards&yand R.& N&et, Phys. Rev. A8 097% 203. 141D.R. Bates, K-L. Bell and A.E. Kingston, Proc. i’hy% SW. (London) 91(1967)
‘.
.,
..,
288.
C!H?3WCAt PHY!XiS
volume 25;numbkQ
I_ETi’EiS
:
15 April 1974
._
DEACTIVATION OF He 2% BY THERMAL ELECTRONS RX. NEiBE’k, R.S. CIBEROI* I&VResearch Labomtory, San Jose. CahfOna 95193. US4 and
P/~ysicsDepartment. University
J.N. BARDSLEY of Pittsburgh,Pittsburglt. Pemtsyhwtia 15213, US.4
Received 7 February 1974
The rate coefficient for deactivation of the metastable 23S state of He by impact of thermal electrons is deduced from recent cakulations of inefastic electron-He cross sections. The deactivation rate is found to be nearly constant with tethperature. Computed values range between 2.82 and 3.03
Cross sections
for electron
impact
excitation
of
Table 1 Total computed deactivation cross section Qzt for H=z$~~S-* 1 ‘S) in units no: and k, Qa, in units nao for dent eIectron wavevector k2(uij*)
the n = 2 states of He have recently been computed by a variational method [I]. The variational method is equivaIent to solution of continuum Bethe-Goldstone equations for the coupled open scattering channels resulting from target atom states lrS, 23S, 2’5, 23P, 2tP, including the most important effects’ of poia~ation or virtual excitation of the n = 2 electrans. The computed 1’S + ?3S excitation cross section is in excellent agreement.with the experimental crosssection f2], if the arbitrary absolute norm&zation of the latter is adjusted to the computed cross section. In view of this agreement,.the corresponding de+xzt%arrbncross sectiop~hould be given to the_ use-’ ful accuracy by the variagorial caIcu!atitins. T&e purpose of this note is to present this d&a, which cann& be.deduced’accurateIy from the graphs published in ref. $ I], and t& derive from it the rate con&ant KB,(2, I) for .deactiv&tion of He(2%) by thermal.- .. electrons. By d&i&d balance, the dea&iv&ion cross &&on
k2
O-02
7.968
0.1594
0.14 0.18 0.22
O-i385 0.1542 0.1763 0.1672 0.1310
0.26 Q.30 0.34
0.319 0.282 -0.180
0.0828 0.0845 0.0611
0.38 0.42
0.x45 0.1 I?
0.0493
Q;; =
;
0.10
-.
kzQzt
2.308 1.542 1.260 0.929 ‘_. 0596
0.06
..
Q21
inci-
0.055 I
0)
where, in Hartree atomic units, the electron energy in the 11s channel iiki, and the energy in the 23S
channcI~is&k$.For low-energy eIeetrons in the second channel, analysis of threshold behavior (3 indicates by, :.:.:.. ‘that Qi;? should va& as kz andQ21 as kg 1 , and that . .._’ : -..._., ._-., ’ the Cross s&+ons are dqminated by the s-wave (2s ‘_ _ . 2.gcatt&pg state&.tiomputed values of Qzr are shown ’ *‘s’&or@d’in pa by-t% dffioe of Na& R&ear& &es : ‘.. .. : in table 1..,Asexpected;&2 Q2i is nearly constant, alC+rtict No_~NOOi&720051; -. .. : ‘. . .; _ ‘. Qzl. is related t+the 2% excitation cross sectiWQ12-
‘. .,. .
-.
;
..-, ..
1 . .
-. _.:
” I ‘:,
-.
.’
_.
._‘..“.
.*
;
,.
.:
:.
., .
.
.,
-‘-
_,.
.‘.
.
,-.
.”
‘..I
,:::-
*
.I
.;
.
_.. ,?. ,_ .-.’::,
. .._,.- _. .. _ -’ __ : ..;:
,, ., ., ,__.. .::-.::.
-..‘.
. .
1, :-:.; ‘; _:_:; _::__. -;.:, _’ _:..:‘-:_.. .. ._
587 _ ,c.’ .,. ( ,. -_.
:
CHEMIQIL PHYSICS LETTERS though
the zP contribution,
which increases
with-k:,
Tablei Deactivation rate coefficient for He(23S -, I’S)
is significant for k2 > 0.10 a&‘. The % contribution to k2 = 0.22 a;’ is less than lO%, and higher partial waves have not been included in the calculations. The deactivation rate coefficient KH,(2, 1) is the .thermal average of u& , where u is the electron Velocity. In atomic units (n(yca%), this is just the thermal average of k, QZ1, K(2,l)
=
Jr exp(-k2/2kg
T)k2dk.
’
42 379 1053
(2)
variable
t* = k;/2kB T _ is
k2Q21 =g(t)=g,,
+g2t2 +g4t4 +.__,
e-r%2n+2dt
= +r(n
+$)
=g0’+3g2
+ ag4
(6)
For numerical values ofg(t), eq. (2) can be evaluated by Gauss quadratilre, using t2ePt2 as the density function in the integrals. For an IV-point quadrature, the weights toi and abscissae ti are to be chosen so that
T= k;/3kB
F. wi8ttf> ‘
(7)
gives the exact value of K(2,l)
for any polynomial including tkms up to g4N_2 _The level of approximation. that neglects terms beyond g2 corresponds to N= 1, with : r2=3-.. w1=l, 1. 1 @I In order to:deter&ne definitive~values of the deexcitation cross section;.it.wckld be necessary tb carry put izal&lations on-a more cIos&ly spaced grid ofk2 %%s HXW thy= i+labl~
;_.,_.
.:
_’ _-..__., ..-_.-.;.,.- . . , ;
2.52
bye, yd to refine-the
_.. .~.-. ,.;. .b _._..y_.--__ ::: _:.,.-, ‘.‘ :.I._-,_. -- .; -_ ._
2.94 2.83 2.82 2.94 3.03
calculations
until k2 Q21 becomes
insensi-
tive to further improvements of the variational wavetinction. Since this has not been done in the present case, a numerical or least-squares fit to k2 Q21 may not be reliable to terms beyondg2_ If this is true, it would be illusory to use a quadrature formula more elaborate than the one-point formula with constants given by eq. (8)_ Since WI = 1, the one-point formula simply associates each given value of k2 Qzl with K(2,l) for an effective temperature determined by eq. (3) with t2 = $. This is
(5)
+ .. . .
5095
variational
and K(2,l)
2.97 3.39 3.22
(4)
the integrals required are of the form
s 0
2063 3410
125 250 500 1000 2000
(3)
If this polynomial
:
3.07 2.67
Table 3 Deactivation rate coefficient (units 10mgcm3/sec), obtained by thermal average of kQ values given in table 1
where kB is the Bokzmann constant (3.1667 X low6 au/%)_ From the values given in table 1 it appears reasonable to approximate k2 Q2i by a polynomial in k$ or in the dimensionless
in units
10-9cm3/sec
T) [kQzl ] k*dk
I,” exp(-k2/2k,
.lS A~+1974
_
The resulting values of K(2, l), for effective values of T corresponding to the entries in table 1, are listed in table % It should be noted that es._(g) is just the: &s&al equipartition law for the momentum of free electrons. However, the. a&ume@ here is riot based on’any assumption of sha~$y-determiried electron mom&turn,but rather on the relatively smooth behavior of k2 Qzl over a range qf en@rw.$ahies. tint3 more-detailed ; .: _- c?mputationi .ofk2 Qil have tieen darried out, it is ., felt *at the pre$entaGoti df yaluespf K(2,.l)+ giy& assump-._:. : -. ,.._.intable _.. 2.k t$e best way to a&id .. I, arbitnky .. .,__ :: _: .:.:
: ..-.: 1.‘.:,
j
_ .I -_ : .:. -..-.. _.,‘. _,- ._.-. -:: 7. ._ .... . : .;
,J’ ; _.,..: -;- ,:,.:~L :, .-.:. ‘._-’
., _
‘V&i&e is, number 4 ..‘. 1 :_
t&s
‘,
I
: _.
:
..
;
,’
CfiEFpUqPIWSICS
I’.-.
about tb6 functional pehavio? of ki &ii &I the
energy tinge ctinsidered her& A. -. ‘. The value$‘ofkz& iisted in table.1 ‘de&i a poly-’ -;nomial, ati in eq. (4), in@iding' teinis’up t6gzo. The i_ weights wi in the corresponding equat-int&al quadiaL ture formula a& easily obtained by inverting m i I X 1I m&ix. Values of K(2, I) cdtiputed by this quadrature formula are given in table 3.
In com~a~son with values 0fk”~o(2,1) obtained by B&es et al. [4], from e&lie1 scattering .d@a, the
_.
I&lTERS
.’
...
‘.
__
.._ .:
_.
.‘_’
.,
:’
..,‘.
:
,,
:.
._,
:
I5 April 1974
present vat&s ar~&&ficantly larger at tow tempera‘.tures, and are eisenti~y c&xtant with te@perature.
: .. ‘, .: ,: :- .‘,_, : : -, Refeiexices
:’ ,:, .,
[l] R.$, o&&i and RX. Nesbet, Phys. Rev. AS (1973) 2969. 12J H.H. Bron@rsma,F.W.E.Knoop and C. Back%Chem. Phys. Letters 13 (1972) 16.
f3] J.N. Bards&yand R.& N&et, Phys. Rev. A8 097% 203. 141D.R. Bates, K-L. Bell and A.E. Kingston, Proc. i’hy% SW. (London) 91(1967)
‘.
.,
..,
288.